On Cusick-Cheon's Conjecture About Balanced Boolean Functions in the Cosets of the Binary Reed-Muller Code

TL;DR

This paper proves an extension of Cusick-Cheon's conjecture regarding balanced Boolean functions within cosets of binary Reed-Muller codes for specific code parameters, advancing understanding in coding theory and Boolean function analysis.

Contribution

It provides a proof of an amplified form of Cusick-Cheon's conjecture for certain Reed-Muller code parameters, specifically when k=1 or k≥(m-1)/2.

Findings

Confirmed the conjecture for k=1

Confirmed the conjecture for k≥(m-1)/2

Extended the understanding of balanced Boolean functions in Reed-Muller codes

Abstract

It is proved an amplification of Cusick-Cheon's conjecture on balanced Boolean functions in the cosets of the binary Reed-Muller code RM(k,m) of order k and length 2^m, in the cases where k = 1 or k >= (m-1)/2.